System and method for correlating point-to-point sky clearness for use in photovoltaic fleet output estimation with the aid of a digital computer

ABSTRACT

Statistically representing point-to-point photovoltaic power estimation and area-to-point conversion of satellite pixel irradiance data are described. Accuracy on correlated overhead sky clearness is bounded by evaluating a mean and standard deviation between recorded irradiance measures and the forecast irradiance measures. Sky clearness over the two locations is related with a correlation coefficient by solving an empirically-derived exponential function of the temporal distance. Each forecast clearness index is weighted by the correlation coefficient to form an output set of forecast clearness indexes and the mean and standard deviation are proportioned. Additionally, accuracy on correlated satellite imagery is bounded by converting collective irradiance into point clearness indexes. A mean and standard deviation for the point clearness indexes is evaluated. The mean is set as an area clearness index for the bounded area. For each point, a variance of the point clearness index is determined and the mean and standard deviation are proportioned.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application is a continuation of U.S. Pat. No. 9,285,505,issued Mar. 15, 2016; which is a continuation of U.S. Pat. No.8,577,612, issued Nov. 5, 2013; which is a continuation of U.S. Pat. No.8,437,959, issued May 7, 2013; which is a continuation of U.S. Pat. No.8,335,649, issued Dec. 18, 2012; which is a continuation of U.S. Pat.No. 8,165,812, issued Apr. 24, 2012, the priority dates of which areclaimed and the disclosures of which are incorporated by reference.

This invention was made with State of California support under AgreementNumber 722. The California Public Utilities Commission of the State ofCalifornia has certain rights to this invention.

FIELD

This application relates in general to photovoltaic power generationfleet planning and operation and, in particular, to a system and methodfor correlating point-to-point sky clearness for use in photovoltaicfleet output estimation with the aid of a digital computer.

BACKGROUND

The manufacture and usage of photovoltaic systems has advancedsignificantly in recent years due to a continually growing demand forrenewable energy resources. The cost per watt of electricity generatedby photovoltaic systems has decreased more dramatically, especially whencombined with government incentives offered to encourage photovoltaicpower generation. Photovoltaic systems are widely applicable asstandalone off-grid power systems, sources of supplemental electricity,such as for use in a building or house, and as power grid-connectedsystems. Typically, when integrated into a power grid, photovoltaicsystems are collectively operated as a fleet, although the individualsystems in the fleet may be deployed at different physical locationswithin a geographic region.

Grid connection of photovoltaic power generation fleets is a fairlyrecent development. In the United States, the Energy Policy Act of 1992deregulated power utilities and mandated the opening of access to powergrids to outsiders, including independent power providers, electricityretailers, integrated energy companies, and Independent System Operators(ISOs) and Regional Transmission Organizations (RTOs). A power grid isan electricity generation, transmission, and distribution infrastructurethat delivers electricity from supplies to consumers. As electricity isconsumed almost immediately upon production, power generation andconsumption must be balanced across the entire power grid. A large powerfailure in one part of the grid could cause electrical current toreroute from remaining power generators over transmission lines ofinsufficient capacity, which creates the possibility of cascadingfailures and widespread power outages.

As a result, both planners and operators of power grids need to be ableto accurately gauge on-going power generation and consumption, andphotovoltaic fleets participating as part of a power grid are expectedto exhibit predictable power generation behaviors. Power production datais needed at all levels of a power grid to which a photovoltaic fleet isconnected, especially in smart grid integration, as well as by operatorsof distribution channels, power utilities, ISOs, and RTOs. Photovoltaicfleet power production data is particularly crucial where a fleet makesa significant contribution to the grid's overall energy mix.

A grid-connected photovoltaic fleet could be dispersed over aneighborhood, utility region, or several states and its constituentphotovoltaic systems could be concentrated together or spread out.Regardless, the aggregate grid power contribution of a photovoltaicfleet is determined as a function of the individual power contributionsof its constituent photovoltaic systems, which in turn, may havedifferent system configurations and power capacities. The systemconfigurations may vary based on operational features, such as size andnumber of photovoltaic arrays, the use of fixed or tracking arrays,whether the arrays are tilted at different angles of elevation or areoriented along differing azimuthal angles, and the degree to which eachsystem is covered by shade due to clouds.

Photovoltaic system power output is particularly sensitive to shadingdue to cloud cover, and a photovoltaic array with only a small portioncovered in shade can suffer a dramatic decrease in power output. For asingle photovoltaic system, power capacity is measured by the maximumpower output determined under standard test conditions and is expressedin units of Watt peak (Wp). However, at any given time, the actual powercould vary from the rated system power capacity depending upongeographic location, time of day, weather conditions, and other factors.Moreover, photovoltaic fleets with individual systems scattered over alarge geographical area are subject to different location-specific cloudconditions with a consequential affect on aggregate power output.

Consequently, photovoltaic fleets operating under cloudy conditions canexhibit variable and unpredictable performance. Conventionally, fleetvariability is determined by collecting and feeding direct powermeasurements from individual photovoltaic systems or equivalentindirectly derived power measurements into a centralized controlcomputer or similar arrangement. To be of optimal usefulness, the directpower measurement data must be collected in near real time at finegrained time intervals to enable a high resolution time series of poweroutput to be created. However, the practicality of such an approachdiminishes as the number of systems, variations in systemconfigurations, and geographic dispersion of the photovoltaic fleetgrow. Moreover, the costs and feasibility of providing remote powermeasurement data can make high speed data collection and analysisinsurmountable due to the bandwidth needed to transmit and the storagespace needed to contain collected measurements, and the processingresources needed to scale quantitative power measurement analysisupwards as the fleet size grows.

For instance, one direct approach to obtaining high speed time seriespower production data from a fleet of existing photovoltaic systems isto install physical meters on every photovoltaic system, record theelectrical power output at a desired time interval, such as every 10seconds, and sum the recorded output across all photovoltaic systems inthe fleet at each time interval. The totalized power data from thephotovoltaic fleet could then be used to calculate the time-averagedfleet power, variance of fleet power, and similar values for the rate ofchange of fleet power. An equivalent direct approach to obtaining highspeed time series power production data for a future photovoltaic fleetor an existing photovoltaic fleet with incomplete metering and telemetryis to collect solar irradiance data from a dense network of weathermonitoring stations covering all anticipated locations of interest atthe desired time interval, use a photovoltaic performance model tosimulate the high speed time series output data for each photovoltaicsystem individually, and then sum the results at each time interval.

With either direct approach, several difficulties arise. First, in termsof physical plant, calibrating, installing, operating, and maintainingmeters and weather stations is expensive and detracts from cost savingsotherwise afforded through a renewable energy source. Similarly,collecting, validating, transmitting, and storing high speed data forevery photovoltaic system or location requires collateral datacommunications and processing infrastructure, again at possiblysignificant expense. Moreover, data loss occurs whenever instrumentationor data communications do not operate reliably.

Second, in terms of inherent limitations, both direct approaches onlywork for times, locations, and photovoltaic system configurations whenand where meters are pre-installed; thus, high speed time series powerproduction data is unavailable for all other locations, time periods,and photovoltaic system configurations. Both direct approaches alsocannot be used to directly forecast future photovoltaic systemperformance since meters must be physically present at the time andlocation of interest. Fundamentally, data also must be recorded at thetime resolution that corresponds to the desired output time resolution.While low time-resolution results can be calculated from high resolutiondata, the opposite calculation is not possible. For example,photovoltaic fleet behavior with a 10-second resolution cannot bedetermined from data collected by existing utility meters that collectthe data with a 15-minute resolution.

The few solar data networks that exist in the United States, such as theARM network, described in G. M. Stokes et al., “The atmosphericradiation measurement (ARM) program: programmatic background and designof the cloud and radiation test bed,” Bulletin of Am. MeteorologicalSociety 75, 1201-1221 (1994), the disclosure of which is incorporated byreference, and the SURFRAD network, do not have high density networks(the closest pair of stations in the ARM network is 50 km apart) norhave they been collecting data at a fast rate (the fastest rate is 20seconds at ARM network and one minute at SURFRAD network).

The limitations of the direct measurement approaches have promptedresearchers to evaluate other alternatives. Researchers have installeddense networks of solar monitoring devices in a few limited locations,such as described in S. Kuszamaul et al., “Lanai High-Density IrradianceSensor Network for Characterizing Solar Resource Variability of MW-ScalePV System.” 35^(th) Photovoltaic Specialists Conf., Honolulu, Hi. (Jun.20-25, 2010), and R. George, “Estimating Ramp Rates for Large PV SystemsUsing a Dense Array of Measured Solar Radiation Data,” Am. Solar EnergySociety Annual Conf. Procs., Raleigh, N.C. (May 18, 2011), thedisclosures of which are incorporated by reference. As data are beingcollected, the researchers examine the data to determine if there areunderlying models that can translate results from these devices tophotovoltaic fleet production at a much broader area, yet fail toprovide translation of the data. In addition, half-hour or hourlysatellite irradiance data for specific locations and time periods ofinterest have been combined with randomly selected high speed data froma limited number of ground-based weather stations, such as described inCAISO 2011. “Summary of Preliminary Results of 33% Renewable IntegrationStudy-2010,” Cal. Public Util. Comm. (Apr. 29, 2011) and J. Stein,“Simulation of 1-Minute Power Output from Utility-Scale PhotovoltaicGeneration Systems,” Am. Solar Energy Society Annual Conf. Procs.,Raleigh, N.C. (May 18, 2011), the disclosures of which are incorporatedby reference. This approach, however, does not produce time synchronizedphotovoltaic fleet variability for any particular time period becausethe locations of the ground-based weather stations differ from theactual locations of the fleet. While such results may be useful as inputdata to photovoltaic simulation models for purpose of performing highpenetration photovoltaic studies, they are not designed to produce datathat could be used in grid operational tools.

Therefore, a need remains for an approach to efficiently estimatingpower output of a photovoltaic fleet in the absence of high speed timeseries power production data.

A further need remains for bounding statistical error on point-to-pointphotovoltaic power estimation and on area-to-point conversion ofsatellite pixel irradiance data.

SUMMARY

Statistically representing point-to-point photovoltaic power estimationand area-to-point conversion of satellite pixel irradiance data aredescribed. Accuracy on correlated overhead sky clearness is bounded byevaluating a mean and standard deviation between recorded irradiancemeasures and the forecast irradiance measures. Sky clearness over thetwo locations is related with a correlation coefficient by solving anempirically-derived exponential function of the temporal distance. Eachforecast clearness index is weighted by the correlation coefficient toform an output set of forecast clearness indexes and the mean andstandard deviation are proportioned. Additionally, accuracy oncorrelated satellite imagery is bounded by converting collectiveirradiance into point clearness indexes. A mean and standard deviationfor the point clearness indexes is evaluated. The mean is set as an areaclearness index for the bounded area. For each point, a variance of thepoint clearness index is determined and the mean and standard deviationare proportioned.

One embodiment provides a system and method for correlatingpoint-to-point sky clearness for use in photovoltaic fleet outputestimation with the aid of a digital computer. A temporal distance,including a physical distance between two locations that are each withina geographic region in proportion to cloud speed within the geographicregion, is provided to a computer. The geographic region is suitable foroperation of a photovoltaic power production fleet. A set of irradianceobservations that has been regularly measured for one of the locationswithin the geographic region is also provided to the computer. A set ofinput sky clearness indexes is determined by the computer as a ratio ofeach of the irradiance observations in the set of irradianceobservations and clear sky global horizontal irradiance. A clearnessindex correlation coefficient between the two locations is calculated bythe computer as an empirically-derived function of the temporal distanceand statistical error between the two locations is bound by evaluating amean and standard deviation. The set of input sky clearness indexes isweighted by the computer by the clearness index correlation coefficientto form a set of output sky clearness indexes, which indicates the skyclearness for the other of the locations. The mean and standarddeviation are proportioned to the output set of sky clearness indexes.

Some of the notable elements of this methodology non-exclusivelyinclude:

(1) Employing a fully derived statistical approach to generatinghigh-speed photovoltaic fleet production data;

(2) Using a small sample of input data sources as diverse asground-based weather stations, existing photovoltaic systems, or solardata calculated from satellite images;

(3) Producing results that are usable for any photovoltaic fleetconfiguration;

(4) Supporting any time resolution, even those time resolutions fasterthan the input data collection rate; and

(5) Providing results in a form that is useful and usable by electricpower grid planning and operation tools.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram showing a computer-implemented method forbounding accuracy on correlated overhead sky clearness for use inphotovoltaic fleet output estimation in accordance with one embodiment.

FIG. 2 is a block diagram showing a computer-implemented system forbounding accuracy on correlated overhead sky clearness for use inphotovoltaic fleet output estimation in accordance with one embodiment.

FIG. 3 is a graph depicting, by way of example, ten hours of time seriesirradiance data collected from a ground-based weather station with10-second resolution.

FIG. 4 is a graph depicting, by way of example, the clearness index thatcorresponds to the irradiance data presented in FIG. 3.

FIG. 5 is a graph depicting, by way of example, the change in clearnessindex that corresponds to the clearness index presented in FIG. 4.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5.

FIGS. 7A-7B are photographs showing, by way of example, the locations ofthe Cordelia Junction and Napa high density weather monitoring stations.

FIGS. 8A-8B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds.

FIGS. 9A-9F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance.

FIGS. 10A-10F are graphs depicting, by way of example, the sameinformation as depicted in FIGS. 9A-9F versus temporal distance.

FIGS. 11A-11F are graphs depicting, by way of example, the predictedversus the measured variances of clearness indexes using differentreference time intervals.

FIGS. 12A-12F are graphs depicting, by way of example, the predictedversus the measured variances of change in clearness indexes usingdifferent reference time intervals.

FIGS. 13A-13F are graphs and a diagram depicting, by way of example,application of the methodology described herein to the Napa network.

FIG. 14 is a graph depicting, by way of example, an actual probabilitydistribution for a given distance between two pairs of locations, ascalculated for a 1,000 meter×1,000 meter grid in one square meterincrements.

FIG. 15 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

FIG. 16 is a graph depicting, by way of example, results generated byapplication of Equation (65).

FIG. 17 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions.

FIG. 18 is a graph depicting, by way of example, results by applicationof the model.

DETAILED DESCRIPTION

Photovoltaic cells employ semiconductors exhibiting a photovoltaiceffect to generate direct current electricity through conversion ofsolar irradiance. Within each photovoltaic cell, light photons exciteelectrons in the semiconductors to create a higher state of energy,which acts as a charge carrier for electric current. A photovoltaicsystem uses one or more photovoltaic panels that are linked into anarray to convert sunlight into electricity. In turn, a collection ofphotovoltaic systems can be collectively operated as a photovoltaicfleet when integrated into a power grid, although the constituentphotovoltaic systems may actually be deployed at different physicallocations within a geographic region.

To aid with the planning and operation of photovoltaic fleets, whetherat the power grid, supplemental, or standalone power generation levels,high resolution time series of power output data is needed toefficiently estimate photovoltaic fleet power production. Thevariability of photovoltaic fleet power generation under cloudyconditions can be efficiently estimated, even in the absence of highspeed time series power production data, by applying a fully derivedstatistical approach. FIG. 1 is a flow diagram showing acomputer-implemented method 10 for bounding accuracy on correlatedoverhead sky clearness for use in photovoltaic fleet output estimationin accordance with one embodiment. The method 10 can be implemented insoftware and execution of the software can be performed on a computersystem, such as further described infra, as a series of process ormethod modules or steps.

Preliminarily, a time series of solar irradiance data is obtained (step11) for a set of locations representative of the geographic regionwithin which the photovoltaic fleet is located or intended to operate,as further described infra with reference to FIG. 3. Each time seriescontains solar irradiance observations electronically recorded at knowninput time intervals over successive time periods. The solar irradianceobservations can include irradiance measured by a representative set ofground-based weather stations (step 12), existing photovoltaic systems(step 13), satellite observations (step 14), or some combinationthereof. Other sources of the solar irradiance data are possible.

Next, the solar irradiance data in the time series is converted overeach of the time periods, such as at half-hour intervals, into a set ofclearness indexes, which are calculated relative to clear sky globalhorizontal irradiance. The set of clearness indexes are interpreted intoas irradiance statistics (step 15), as further described infra withreference to FIG. 4-6. The irradiance statistics for each of thelocations is combined into fleet irradiance statistics applicable overthe geographic region of the photovoltaic fleet. A time lag correlationcoefficient for an output time interval is also determined to enable thegeneration of an output time series at any time resolution, even fasterthan the input data collection rate.

Finally, power statistics, including a time series of the powerstatistics for the photovoltaic fleet, are generated (step 17) as afunction of the fleet irradiance statistics and system configuration,particularly the geographic distribution and power rating of thephotovoltaic systems in the fleet (step 16). The resultant high-speedtime series fleet performance data can be used to predictably estimatepower output and photovoltaic fleet variability by fleet planners andoperators, as well as other interested parties.

The calculated irradiance statistics are combined with the photovoltaicfleet configuration to generate the high-speed time series photovoltaicproduction data. In a further embodiment, the foregoing methodology canbe may also require conversion of weather data for a region, such asdata from satellite regions, to average point weather data. Anon-optimized approach would be to calculate a correlation coefficientmatrix on-the-fly for each satellite data point. Alternatively, aconversion factor for performing area-to-point conversion of satelliteimagery data is described in commonly-assigned U.S. Pat. No. 8,165,813,issued Apr. 24, 2012, the disclosure of which is incorporated byreference.

The high resolution time series of power output data is determined inthe context of a photovoltaic fleet, whether for an operational fleetdeployed in the field, by planners considering fleet configuration andoperation, or by other individuals interested in photovoltaic fleetvariability and prediction. FIG. 2 is a block diagram showing acomputer-implemented system 20 for bounding accuracy on correlatedoverhead sky clearness for use in photovoltaic fleet output estimationin accordance with one embodiment. Time series power output data for aphotovoltaic fleet is generated using observed field conditions relatingto overhead sky clearness. Solar irradiance 23 relative to prevailingcloudy conditions 22 in a geographic region of interest is measured.Direct solar irradiance measurements can be collected by ground-basedweather stations 24. Solar irradiance measurements can also be inferredby the actual power output of existing photovoltaic systems 25.Additionally, satellite observations 26 can be obtained for thegeographic region. Both the direct and inferred solar irradiancemeasurements are considered to be sets of point values that relate to aspecific physical location, whereas satellite imagery data is consideredto be a set of area values that need to be converted into point values,as further described infra. Still other sources of solar irradiancemeasurements are possible.

The solar irradiance measurements are centrally collected by a computersystem 21 or equivalent computational device. The computer system 21executes the methodology described supra with reference to FIG. 1 and asfurther detailed herein to generate time series power data 30 and otheranalytics, which can be stored or provided 27 to planners, operators,and other parties for use in solar power generation 28 planning andoperations. The data feeds 29 a-c from the various sources of solarirradiance data need not be high speed connections; rather, the solarirradiance measurements can be obtained at an input data collection rateand application of the methodology described herein provides thegeneration of an output time series at any time resolution, even fasterthan the input time resolution. The computer system 21 includes hardwarecomponents conventionally found in a general purpose programmablecomputing device, such as a central processing unit, memory, userinterfacing means, such as a keyboard, mouse, and display, input/outputports, network interface, and non-volatile storage, and execute softwareprograms structured into routines, functions, and modules for executionon the various systems. In addition, other configurations ofcomputational resources, whether provided as a dedicated system orarranged in client-server or peer-to-peer topologies, and includingunitary or distributed processing, communications, storage, and userinterfacing, are possible.

The detailed steps performed as part of the methodology described suprawith reference to FIG. 1 will now be described.

Obtain Time Series Irradiance Data

The first step is to obtain time series irradiance data fromrepresentative locations. This data can be obtained from ground-basedweather stations, existing photovoltaic systems, a satellite network, orsome combination sources, as well as from other sources. The solarirradiance data is collected from several sample locations across thegeographic region that encompasses the photovoltaic fleet.

Direct irradiance data can be obtained by collecting weather data fromground-based monitoring systems. FIG. 3 is a graph depicting, by way ofexample, ten hours of time series irradiance data collected from aground-based weather station with 10-second resolution, that is, thetime interval equals ten seconds. In the graph, the blue line 32 is themeasured horizontal irradiance and the black line 31 is the calculatedclear sky horizontal irradiance for the location of the weather station.

Irradiance data can also be inferred from select photovoltaic systemsusing their electrical power output measurements. A performance modelfor each photovoltaic system is first identified, and the input solarirradiance corresponding to the power output is determined.

Finally, satellite-based irradiance data can also be used. As satelliteimagery data is pixel-based, the data for the geographic region isprovided as a set of pixels, which span across the region andencompassing the photovoltaic fleet.

Calculate Irradiance Statistics

The time series irradiance data for each location is then converted intotime series clearness index data, which is then used to calculateirradiance statistics, as described infra.

Clearness Index (Kt)

The clearness index (Kt) is calculated for each observation in the dataset. In the case of an irradiance data set, the clearness index isdetermined by dividing the measured global horizontal irradiance by theclear sky global horizontal irradiance, may be obtained from any of avariety of analytical methods. FIG. 4 is a graph depicting, by way ofexample, the clearness index that corresponds to the irradiance datapresented in FIG. 3. Calculation of the clearness index as describedherein is also generally applicable to other expressions of irradianceand cloudy conditions, including global horizontal and direct normalirradiance.

Change in Clearness Index (ΔKt)

The change in clearness index (ΔKt) over a time increment of Δt is thedifference between the clearness index starting at the beginning of atime increment t and the clearness index starting at the beginning of atime increment t, plus a time increment Δt. FIG. 5 is a graph depicting,by way of example, the change in clearness index that corresponds to theclearness index presented in FIG. 4.

Time Period

The time series data set is next divided into time periods, forinstance, from five to sixty minutes, over which statisticalcalculations are performed. The determination of time period is selecteddepending upon the end use of the power output data and the timeresolution of the input data. For example, if fleet variabilitystatistics are to be used to schedule regulation reserves on a 30-minutebasis, the time period could be selected as 30 minutes. The time periodmust be long enough to contain a sufficient number of sampleobservations, as defined by the data time interval, yet be short enoughto be usable in the application of interest. An empirical investigationmay be required to determine the optimal time period as appropriate.

Fundamental Statistics

Table 1 lists the irradiance statistics calculated from time series datafor each time period at each location in the geographic region. Notethat time period and location subscripts are not included for eachstatistic for purposes of notational simplicity.

TABLE 1 Statistic Variable Mean clearness index μ_(Kt) Varianceclearness index σ_(Kt) ² Mean clearness index change μ_(ΔKt) Varianceclearness index change σ_(ΔKt) ²

Table 2 lists sample clearness index time series data and associatedirradiance statistics over five-minute time periods. The data is basedon time series clearness index data that has a one-minute time interval.The analysis was performed over a five-minute time period. Note that theclearness index at 12:06 is only used to calculate the clearness indexchange and not to calculate the irradiance statistics.

TABLE 2 Clearness index (Kt) Clearness index Change (ΔKt) 12:00 50% 40%12:01 90%  0% 12:02 90% −80%  12:03 10%  0% 12:04 10% 80% 12:05 90%−40%  12:06 50% Mean (μ) 57%  0% Variance (σ²) 13% 27%

The mean clearness index change equals the first clearness index in thesucceeding time period, minus the first clearness index in the currenttime period divided by the number of time intervals in the time period.The mean clearness index change equals zero when these two values arethe same. The mean is small when there are a sufficient number of timeintervals. Furthermore, the mean is small relative to the clearnessindex change variance. To simplify the analysis, the mean clearnessindex change is assumed to equal zero for all time periods.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5 using a half-hour hour time period.Note that FIG. 6 presents the standard deviations, determined as thesquare root of the variance, rather than the variances, to present thestandard deviations in terms that are comparable to the mean.

Calculate Fleet Irradiance Statistics

Irradiance statistics were calculated in the previous section for thedata stream at each sample location in the geographic region. Themeaning of these statistics, however, depends upon the data source.Irradiance statistics calculated from a ground-based weather stationdata represent results for a specific geographical location as pointstatistics. Irradiance statistics calculated from satellite datarepresent results for a region as area statistics. For example, if asatellite pixel corresponds to a one square kilometer grid, then theresults represent the irradiance statistics across a physical area onekilometer square.

Average irradiance statistics across the photovoltaic fleet region are acritical part of the methodology described herein. This section presentsthe steps to combine the statistical results for individual locationsand calculate average irradiance statistics for the region as a whole.The steps differs depending upon whether point statistics or areastatistics are used.

Irradiance statistics derived from ground-based sources simply need tobe averaged to form the average irradiance statistics across thephotovoltaic fleet region. Irradiance statistics from satellite sourcesare first converted from irradiance statistics for an area intoirradiance statistics for an average point within the pixel. The averagepoint statistics are then averaged across all satellite pixels todetermine the average across the photovoltaic fleet region.

Mean Clearness Index (μ _(Kt) ) and Mean Change in Clearness Index (μ_(ΔKt) )

The mean clearness index should be averaged no matter what input datasource is used, whether ground, satellite, or photovoltaic systemoriginated data. If there are N locations, then the average clearnessindex across the photovoltaic fleet region is calculated as follows.

$\begin{matrix}{\mu_{\overset{\_}{Kt}} = {\sum\limits_{i = 1}^{N}\;\frac{\mu_{{Kt}_{i}}}{N}}} & (1)\end{matrix}$

The mean change in clearness index for any period is assumed to be zero.As a result, the mean change in clearness index for the region is alsozero.

$\begin{matrix}{\mu_{\overset{\_}{\Delta\;{Kt}}} = 0} & (2)\end{matrix}$

Convert Area Variance to Point Variance

The following calculations are required if satellite data is used as thesource of irradiance data. Satellite observations represent valuesaveraged across the area of the pixel, rather than single pointobservations. The clearness index derived from this data (Kt^(Area)) maytherefore be considered an average of many individual pointmeasurements.

$\begin{matrix}{{Kt}^{Area} = {\sum\limits_{i = 1}^{N}\;\frac{{Kt}^{i}}{N}}} & (3)\end{matrix}$

As a result, the variance of the area clearness index based on satellitedata can be expressed as the variance of the average clearness indexesacross all locations within the satellite pixel.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{{VAR}\left\lbrack {Kt}^{Area} \right\rbrack} = {{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}\;\frac{{Kt}^{i}}{N}} \right\rbrack}}} & (4)\end{matrix}$

The variance of a sum, however, equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;{{COV}\left\lbrack {{Kt}^{i},{Kt}^{j}} \right\rbrack}}}}} & (5)\end{matrix}$

Let ρ^(Kt) ^(i) ^(,Kt) ^(j) represents the correlation coefficientbetween the clearness index at location i and location j within thesatellite pixel. By definition of correlation coefficient, COV[Kt^(i),Kt^(j)]=σ_(Kt) ^(i)σ_(Kt) ^(j)ρ^(Kt) ^(i) ^(,Kt) ^(j) . Furthermore,since the objective is to determine the average point variance acrossthe satellite pixel, the standard deviation at any point within thesatellite pixel can be assumed to be the same and equals σ_(Kt), whichmeans that σ_(Kt) ^(i)σ_(Kt) ^(j)=σ_(Kt) ² for all location pairs. As aresult, COV[Kt^(i), Kt^(j)]=σ_(Kt) ²ρ^(Kt) ^(i) ^(,Kt) ^(j) .Substituting this result into Equation (5) and simplify.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{\sigma_{Kt}^{2}\left( \frac{1}{N^{2}} \right)}{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;\rho^{{Kt}^{i},{Kt}^{j}}}}}} & (6)\end{matrix}$

Suppose that data was available to calculate the correlation coefficientin Equation (6). The computational effort required to perform a doublesummation for many points can be quite large and computationallyresource intensive. For example, a satellite pixel representing a onesquare kilometer area contains one million square meter increments. Withone million increments, Equation (6) would require one trillioncalculations to compute.

The calculation can be simplified by conversion into a continuousprobability density function of distances between location pairs acrossthe pixel and the correlation coefficient for that given distance, asfurther described supra. Thus, the irradiance statistics for a specificsatellite pixel, that is, an area statistic, rather than a pointstatistics, can be converted into the irradiance statistics at anaverage point within that pixel by dividing by a “Area” term (A), whichcorresponds to the area of the satellite pixel. Furthermore, theprobability density function and correlation coefficient functions aregenerally assumed to be the same for all pixels within the fleet region,making the value of A constant for all pixels and reducing thecomputational burden further. Details as to how to calculate A are alsofurther described supra.

$\begin{matrix}{{\sigma_{Kt}^{2} = \frac{\sigma_{{Kt} - {Area}}^{2}}{A_{Kt}^{{Satellite}\mspace{14mu}{Pixel}}}}{{where}\text{:}}} & (7) \\{A_{Kt}^{{Satellite}\mspace{14mu}{Pixel}} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;\rho^{i,j}}}}} & (8)\end{matrix}$

Likewise, the change in clearness index variance across the satelliteregion can also be converted to an average point estimate using asimilar conversion factor, A_(ΔKt) ^(Area).

$\begin{matrix}{\sigma_{\Delta\;{Kt}}^{2} = \frac{\sigma_{{\Delta\;{Kt}} - {Area}}^{2}}{A_{\Delta\;{Kt}}^{{Satellite}\mspace{14mu}{Pixel}}}} & (9)\end{matrix}$

Variance of Clearness Index

$\left( \sigma_{\overset{\_}{Kt}}^{2} \right)$

and Variance or Change in Clearness Index

$\left( {\sigma\frac{2}{\Delta\;{Kt}}} \right)$

At this point, the point statistics (σ_(Kt) ² and σ_(ΔKt) ²) have beendetermined for each of several representative locations within the fleetregion. These values may have been obtained from either ground-basedpoint data or by converting satellite data from area into pointstatistics. If the fleet region is small, the variances calculated ateach location i can be averaged to determine the average point varianceacross the fleet region. If there are N locations, then average varianceof the clearness index across the photovoltaic fleet region iscalculated as follows.

$\begin{matrix}{{\sigma\frac{2}{Kt}} = {\sum\limits_{i = 1}^{N}\frac{\sigma_{{Kt}_{i}}^{2}}{N}}} & (10)\end{matrix}$

Likewise, the variance of the clearness index change is calculated asfollows.

$\begin{matrix}{{\sigma\frac{2}{\Delta\;{Kt}}} = {\sum\limits_{i = 1}^{N}\frac{\sigma_{\Delta\;{Kt}_{i}}^{2}}{N}}} & (11)\end{matrix}$Calculate Fleet Power Statistics

The next step is to calculate photovoltaic fleet power statistics usingthe fleet irradiance statistics, as determined supra, and physicalphotovoltaic fleet configuration data. These fleet power statistics arederived from the irradiance statistics and have the same time period.

The critical photovoltaic fleet performance statistics that are ofinterest are the mean fleet power, the variance of the fleet power, andthe variance of the change in fleet power over the desired time period.As in the case of irradiance statistics, the mean change in fleet poweris assumed to be zero.

Photovoltaic System Power for Single System at Time t

Photovoltaic system power output (kW) is approximately linearly relatedto the AC-rating of the photovoltaic system (R in units of kW_(AC))times plane-of-array irradiance. Plane-of-array irradiance can berepresented by the clearness index over the photovoltaic system (KtPV)times the clear sky global horizontal irradiance times an orientationfactor (O), which both converts global horizontal irradiance toplane-of-array irradiance and has an embedded factor that convertsirradiance from Watts/m² to kW output/kW of rating. Thus, at a specificpoint in time (t), the power output for a single photovoltaic system (n)equals:P_(t) ^(n)=R^(n)O_(t) ^(n)KtPV_(t) ^(n)I_(t) ^(Clear,n)  (12)

The change in power equals the difference in power at two differentpoints in time.ΔP _(t,Δt) ^(n) =R ^(n) O _(t+Δt) ^(n) KtPV _(t+Δt) ^(n) I _(t+Δt)^(Clear,n) −R ^(n) O _(t) ^(n) KtPV _(t) ^(n) I _(t) ^(Clear,n)  (13)

The rating is constant, and over a short time interval, the two clearsky plane-of-array irradiances are approximately the same (O_(t+Δ) ^(n)I _(t+Δt) ^(Clear,n)≈O_(t) ^(n)I_(t) ^(Clear,n)), so that the threeterms can be factored out and the change in the clearness index remains.ΔP_(t,Δ) ^(n)≈R^(n)O_(t) ^(n)I_(t) ^(Clear,n)ΔKtPV_(t) ^(n)  (14)

Time Series Photovoltaic Power for Single System

P^(n) is a random variable that summarizes the power for a singlephotovoltaic system n over a set of times for a given time interval andset of time periods. ΔP^(n) is a random variable that summarizes thechange in power over the same set of times.

Mean Fleet Power (μ_(P))

The mean power for the fleet of photovoltaic systems over the timeperiod equals the expected value of the sum of the power output from allof the photovoltaic systems in the fleet.

$\begin{matrix}{\mu_{P} = {E\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{n}I^{{Clear},n}}} \right\rbrack}} & (15)\end{matrix}$

If the time period is short and the region small, the clear skyirradiance does not change much and can be factored out of theexpectation.

$\begin{matrix}{\mu_{P} = {\mu_{I^{Clear}}{E\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{n}}} \right\rbrack}}} & (16)\end{matrix}$

Again, if the time period is short and the region small, the clearnessindex can be averaged across the photovoltaic fleet region and any givenorientation factor can be assumed to be a constant within the timeperiod. The result is that:μ_(P)=R^(Adj.Fleet) μ_(l) _(Clear) μ _(Kt)   (17)where μ_(l)Clear is calculated, μ _(Kt) is taken from Equation (1) and:

$\begin{matrix}{R^{{Adj} \cdot {Fleet}} = {\sum\limits_{n = 1}^{N}{R^{n}O^{n}}}} & (18)\end{matrix}$

This value can also be expressed as the average power during clear skyconditions times the average clearness index across the region.μ_(P)=μ_(P) _(Clear) μ _(Kt)   (19)

Variance of Fleet Power (σ_(P) ²)

The variance of the power from the photovoltaic fleet equals:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{n}I^{{Clear},n}}} \right\rbrack}} & (20)\end{matrix}$

If the clear sky irradiance is the same for all systems, which will bethe case when the region is small and the time period is short, then:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {I^{Clear}{\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{n}}}} \right\rbrack}} & (21)\end{matrix}$

The variance of a product of two independent random variables X, Y, thatis, VAR[XY]) equals E[X]²VAR[Y]+E[Y]²VAR[X]+VAR[X]VAR[Y]. If the Xrandom variable has a large mean and small variance relative to theother terms, then VAR[XV]≈E[X]²VAR[Y]. Thus, the clear sky irradiancecan be factored out of Equation (21) and can be written as:

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}{{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}{KtPV}^{n}O^{n}}} \right\rbrack}}} & (22)\end{matrix}$

The variance of a sum equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{{COV}\left\lbrack {{R^{i}{KtPV}^{i}O^{i}},{R^{j}{KtPV}^{j}O^{j}}} \right\rbrack}}} \right)}} & (23)\end{matrix}$

In addition, over a short time period, the factor to convert from clearsky GHI to clear sky POA does not vary much and becomes a constant. Allfour variables can be factored out of the covariance equation.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right){{COV}\left\lbrack {{KtPV}^{i},{KtPV}^{j}} \right\rbrack}}}} \right)}} & (24)\end{matrix}$

For any i and j, COV[KtPV^(i),KtPV^(j)]=√{square root over (σ_(KtPV)_(i) ²σ_(KtPV) _(j) )}²ρ^(Kt) ^(i) ^(,Kt) ^(j) .

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right)\sqrt{\sigma_{{KtPV}^{i}}^{2}\sigma_{{KtPV}^{j}}^{2}}\rho^{{Kt}^{i},{Kt}^{j}}}}} \right)}} & (25)\end{matrix}$

As discussed supra, the variance of the satellite data required aconversion from the satellite area, that is, the area covered by apixel, to an average point within the satellite area. In the same way,assuming a uniform clearness index across the region of the photovoltaicplant, the variance of the clearness index across a region the size ofthe photovoltaic plant within the fleet also needs to be adjusted. Thesame approach that was used to adjust the satellite clearness index canbe used to adjust the photovoltaic clearness index. Thus, each varianceneeds to be adjusted to reflect the area that the i^(th) photovoltaicplant covers.σ_(KtPV) _(i) ²=A_(Kt) ^(i)σ _(Kt) ²   (26)

Substituting and then factoring the clearness index variance given theassumption that the average variance is constant across the regionyields:

$\begin{matrix}{\sigma_{P}^{2} = {\left( {R^{{Adj} \cdot {Fleet}}\mu_{I^{{Clear}\;}}} \right)^{2}P^{Kt}\sigma\frac{2}{Kt}}} & (27)\end{matrix}$where the correlation matrix equals:

$\begin{matrix}{P^{Kt} = \frac{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}A_{Kt}^{i}} \right)\left( {R^{j}O^{j}A_{Kt}^{j}} \right)\rho^{{Kt}^{i},{Kt}^{j}}}}}{\left( {\sum\limits_{n = 1}^{N}{R^{n}O^{n}}} \right)^{2}}} & (28)\end{matrix}$

R^(Adj.Fleet) μ_(I)Clear in Equation (27) can be written as the powerproduced by the photovoltaic fleet under clear sky conditions, that is:

$\begin{matrix}{\sigma_{P}^{2} = {\mu_{P^{Clear}}^{2}P^{\;{K\; t}}\sigma\frac{2}{Kt}}} & (29)\end{matrix}$

If the region is large and the clearness index mean or variances varysubstantially across the region, then the simplifications may not beable to be applied. Notwithstanding, if the simplification isinapplicable, the systems are likely located far enough away from eachother, so as to be independent. In that case, the correlationcoefficients between plants in different regions would be zero, so mostof the terms in the summation are also zero and an inter-regionalsimplification can be made. The variance and mean then become theweighted average values based on regional photovoltaic capacity andorientation.

Discussion

In Equation (28), the correlation matrix term embeds the effect ofintra-plant and inter-plant geographic diversification. The area-relatedterms (A) inside the summations reflect the intra-plant power smoothingthat takes place in a large plant and may be calculated using thesimplified relationship, as further discussed supra. These terms arethen weighted by the effective plant output at the time, that is, therating adjusted for orientation. The multiplication of these terms withthe correlation coefficients reflects the inter-plant smoothing due tothe separation of photovoltaic systems from one another.

Variance of Change in Fleet Power (σ_(ΔP) ²)

A similar approach can be used to show that the variance of the changein power equals:

$\begin{matrix}{{\sigma_{\Delta\; P}^{2} = {\mu_{P^{Clear}}^{2}P^{\Delta\;{Kt}}\sigma\frac{2}{\Delta\;{Kt}}}}{{where}\text{:}}} & (30) \\{P^{\Delta\;{Kt}} = \frac{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}A_{\Delta\;{Kt}}^{i}} \right)\left( {R^{j}O^{j}A_{\Delta\;{Kt}}^{j}} \right)\rho^{{\Delta\;{Kt}^{i}},{\Delta\;{Kt}^{j}}}}}}{\left( {\sum\limits_{n = 1}^{N}{R^{n}O^{n}}} \right)^{2}}} & (31)\end{matrix}$

The determination of Equations (30) and (31) becomes computationallyintensive as the network of points becomes large. For example, a networkwith 10,000 photovoltaic systems would require the computation of acorrelation coefficient matrix with 100 million calculations. Thecomputational burden can be reduced in two ways. First, many of theterms in the matrix are zero because the photovoltaic systems arelocated too far away from each other. Thus, the double summation portionof the calculation can be simplified to eliminate zero values based ondistance between locations by construction of a grid of points. Second,once the simplification has been made, rather than calculating thematrix on-the-fly for every time period, the matrix can be calculatedonce at the beginning of the analysis for a variety of cloud speedconditions, and then the analysis would simply require a lookup of theappropriate value.

Time Lag Correlation Coefficient

The next step is to adjust the photovoltaic fleet power statistics fromthe input time interval to the desired output time interval. Forexample, the time series data may have been collected and stored every60 seconds. The user of the results, however, may want to havephotovoltaic fleet power statistics at a 10-second rate. This adjustmentis made using the time lag correlation coefficient. The time lagcorrelation coefficient reflects the relationship between fleet powerand that same fleet power starting one time interval (Δt) later.Specifically, the time lag correlation coefficient is defined asfollows:

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = \frac{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{{P\;}^{\Delta\; t}}^{2}}}} & (32)\end{matrix}$

The assumption that the mean clearness index change equals zero impliesthat σ_(P) _(Δt) ²=σ_(P) ². Given a non-zero variance of power, thisassumption can also be used to show that

$\frac{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}{\sigma_{P}^{2}} = {1 - {\frac{\sigma_{\Delta\; P}^{2}}{2\sigma_{P}^{2}}.}}$Therefore:

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = {1 - \frac{\sigma_{\Delta\; P}^{2}}{2\sigma_{P}^{2}}}} & (33)\end{matrix}$

This relationship illustrates how the time lag correlation coefficientfor the time interval associated with the data collection rate iscompletely defined in terms of fleet power statistics alreadycalculated. A more detailed derivation is described infra.

Equation (33) can be stated completely in terms of the photovoltaicfleet configuration and the fleet region clearness index statistics bysubstituting Equations (29) and (30). Specifically, the time lagcorrelation coefficient can be stated entirely in terms of photovoltaicfleet configuration, the variance of the clearness index, and thevariance of the change in the clearness index associated with the timeincrement of the input data.

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = {1 - \frac{P^{\Delta\;{Kt}}\sigma\frac{2}{\Delta\;{Kt}}}{2P^{Kt}\sigma\frac{2}{Kt}}}} & (34)\end{matrix}$Generate High-Speed Time Series Photovoltaic Fleet Power

The final step is to generate high-speed time series photovoltaic fleetpower data based on irradiance statistics, photovoltaic fleetconfiguration, and the time lag correlation coefficient. This step is toconstruct time series photovoltaic fleet production from statisticalmeasures over the desired time period, for instance, at half-hour outputintervals.

A joint probability distribution function is required for this step. Thebivariate probability density function of two unit normal randomvariables (X and Y) with a correlation coefficient of ρ equals:

$\begin{matrix}{{f\left( {x,y} \right)} = {\frac{1}{2\pi\sqrt{1 - \rho^{2}}}{\exp\left\lbrack {- \frac{\left( {x^{2} + y^{2} - {2\rho\;{xy}}} \right)}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (35)\end{matrix}$

The single variable probability density function for a unit normalrandom variable X alone is

$(x) = {\frac{1}{\sqrt{2\pi}}{{\exp\left( {- \frac{x^{2}}{x}} \right)}.}}$In addition, a conditional distribution for y can be calculated based ona known x by dividing the bivariate probability density function by thesingle variable probability density, that is,

${f\left( {y❘x} \right)} = {\frac{f\left( {x,y} \right)}{f(x)}.}$Making the appropriate substitutions, the result is that the conditionaldistribution of y based on a known x equals:

$\begin{matrix}{{f\left( {y❘x} \right)} = {\frac{1}{\sqrt{2\pi}\sqrt{1 - \rho^{2}}}{\exp\left\lbrack {- \frac{\left( {y - {\rho\; x}} \right)^{2}}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (36)\end{matrix}$

Define a random variable

$Z = \frac{Y - {\rho\; x}}{\sqrt{1 - \rho^{2}}}$and substitute into Equation (36). The result is that the conditionalprobability of z given a known x equals:

$\begin{matrix}{{f\left( {z❘x} \right)} = {\frac{1}{\sqrt{2\pi}}{\exp\left( {- \frac{z^{2}}{2}} \right)}}} & (37)\end{matrix}$

The cumulative distribution function for Z can be denoted by Φ(z*),where z* represents a specific value for z. The result equals aprobability (p) that ranges between 0 (when z*=−∞) and 1 (when z*=∞).The function represents the cumulative probability that any value of zis less than z*, as determined by a computer program or value lookup.

$\begin{matrix}{p = {{\Phi\left( z^{*} \right)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{z^{*}}{{\exp\left( {- \frac{z^{2}}{2}} \right)}\ d\; z}}}}} & (38)\end{matrix}$

Rather than selecting z*, however, a probability p falling between 0 and1 can be selected and the corresponding z* that results in thisprobability found, which can be accomplished by taking the inverse ofthe cumulative distribution function.Φ⁻¹(p)=z*  (39)

Substituting back for z as defined above results in:

$\begin{matrix}{{\Phi^{- 1}(p)} = \frac{y - {\rho\; x}}{\sqrt{1 - \rho^{2}}}} & (40)\end{matrix}$

Now, let the random variables equal

${X = {{\frac{P - \mu_{P}}{\sigma_{P}}{and}\mspace{14mu} Y} = \frac{P^{\Delta\; t} - \mu_{P^{\Delta\; t}}}{\sigma_{P^{\Delta\; t}}}}},$with the correlation coefficient being the time lag correlationcoefficient between P and P^(Δt), that is, let ρ=ρ^(P,P) ^(Δt) . When Δtis small, then the mean and standard deviations for P^(Δt) areapproximately equal to the mean and standard deviation for P. Thus, Ycan be restated as

$Y \approx {\frac{P^{\Delta\; t} - \mu_{P}}{\sigma_{P}}.}$Add a time subscript to all of the relevant data to represent a specificpoint in time and substitute x, y, and ρ into Equation (40).

$\begin{matrix}{{\Phi^{- 1}(p)} = \frac{\left( \frac{P_{t}^{\Delta\; t} - \mu_{P}}{\sigma_{P}} \right) - {\rho^{P,P^{\Delta\; t}}\left( \frac{P_{t} - \mu_{P}}{\sigma_{P}} \right)}}{\sqrt{1 - \rho^{P,P^{\Delta\; t^{2}}}}}} & (41)\end{matrix}$

The random variable P^(Δt), however, is simply the random variable Pshifted in time by a time interval of Δt. As a result, at any given timet, P^(Δt) _(t)=P_(t+Δt). Make this substitution into Equation (41) andsolve in terms of P_(t+Δt).

$\begin{matrix}{P_{t + {\Delta\; t}} = {{\rho^{P,P^{\Delta\; t}}P_{t}} + {\left( {1 - \rho^{P,P^{\Delta\; t}}} \right)\mu_{P}} + {\sqrt{\sigma_{P}^{2}\left( {1 - \rho^{P,P^{\Delta\; t^{2}}}} \right)}{\Phi^{- 1}(p)}}}} & (42)\end{matrix}$

At any given time, photovoltaic fleet power equals photovoltaic fleetpower under clear sky conditions times the average regional clearnessindex, that is, P_(t)=P_(t) ^(Clear) Kt_(t). In addition, over a shorttime period, μ_(P)≈P_(t) ^(Clear) μ _(Kt) and

$\sigma_{P}^{2} \approx {\left( P_{t}^{Clear} \right)^{2}\mspace{14mu} P^{Kt}{\sigma_{\overset{\_}{Kt}}^{2}.}}$Substitute these three relationships into Equation (42) and factor outphotovoltaic fleet power under clear sky conditions (P_(t) ^(Clear)) ascommon to all three terms.

$\begin{matrix}{P_{t + {\Delta\; t}} = {P_{t}^{Clear}\left\lbrack {{\rho^{P,P^{\Delta\; t}}{Kt}_{t}} + {\left( {1 - \rho^{P,P^{\Delta\; t}}} \right)\mu_{\overset{\_}{Kt}}} + {\sqrt{P^{Kt}{\sigma_{\overset{\_}{Kt}}^{2}\left( {1 - \rho^{P,P^{\Delta\; t^{2}}}} \right)}}{\Phi^{- 1}\left( p_{t} \right)}}} \right\rbrack}} & (43)\end{matrix}$

Equation (43) provides an iterative method to generate high-speed timeseries photovoltaic production data for a fleet of photovoltaic systems.At each time step (t+Δt), the power delivered by the fleet ofphotovoltaic systems (P_(t+Δt)) is calculated using input values fromtime step t. Thus, a time series of power outputs can be created. Theinputs include:

-   -   P_(t) ^(Clear)—photovoltaic fleet power during clear sky        conditions calculated using a photovoltaic simulation program        and clear sky irradiance.    -   Kt_(t)—average regional clearness index inferred based on P_(t)        calculated in time step t, that is, Kt_(t)=P_(t)/P_(t) ^(Clear).    -   μ _(Kt) —mean clearness index calculated using time series        irradiance data and Equation (1).

$\sigma_{\overset{\_}{Kt}}^{2}$—variance of the clearness index calculated using time series irradiancedata and Equation (10).

-   -   ρ^(P,P) ^(Δt) —fleet configuration as reflected in the time lag        correlation coefficient calculated using Equation (34). In turn,        Equation (34), relies upon correlation coefficients from        Equations (28) and (31). A method to obtain these correlation        coefficients by empirical means is described in        commonly-assigned U.S. Pat. No. 8,165,811, issued Apr. 24, 2012,        and U.S. Pat. No. 8,165,813, issued Apr. 24, 2012, the        disclosure of which is incorporated by reference.    -   P^(Kt)—fleet configuration as reflected in the clearness index        correlation coefficient matrix calculated using Equation (28)        where, again, the correlation coefficients may be obtained using        the empirical results as further described infra.    -   Φ⁻¹(p_(t))—the inverse cumulative normal distribution function        based on a random variable between 0 and 1.        Derivation of Empirical Models

The previous section developed the mathematical relationships used tocalculate irradiance and power statistics for the region associated witha photovoltaic fleet. The relationships between Equations (8), (28),(31), and (34) depend upon the ability to obtain point-to-pointcorrelation coefficients. This section presents empirically-derivedmodels that can be used to determine the value of the coefficients forthis purpose.

A mobile network of 25 weather monitoring devices was deployed in a 400meter by 400 meter grid in Cordelia Junction, Calif., between Nov. 6,2010, and Nov. 15, 2010, and in a 4,000 meter by 4,000 meter grid inNapa, Calif., between Nov. 19, 2010, and Nov. 24, 2010. FIGS. 7A-7B arephotographs showing, by way of example, the locations of the CordeliaJunction and Napa high density weather monitoring stations.

An analysis was performed by examining results from Napa and CordeliaJunction using 10, 30, 60, 120 and 180 second time intervals over eachhalf-hour time period in the data set. The variance of the clearnessindex and the variance of the change in clearness index were calculatedfor each of the 25 locations for each of the two networks. In addition,the clearness index correlation coefficient and the change in clearnessindex correlation coefficient for each of the 625 possible pairs, 300 ofwhich are unique, for each of the two locations were calculated.

An empirical model is proposed as part of the methodology describedherein to estimate the correlation coefficient of the clearness indexand change in clearness index between any two points by using as inputsthe following: distance between the two points, cloud speed, and timeinterval. For the analysis, distances were measured, cloud speed wasimplied, and a time interval was selected.

The empirical models infra describe correlation coefficients between twopoints (i and j), making use of “temporal distance,” defined as thephysical distance (meters) between points i and j, divided by theregional cloud speed (meters per second) and having units of seconds.The temporal distance answers the question, “How much time is needed tospan two locations?”

Cloud speed was estimated to be six meters per second. Results indicatethat the clearness index correlation coefficient between the twolocations closely matches the estimated value as calculated using thefollowing empirical model:ρ^(Kt) ^(i) ^(,Kt) ^(j) =exp(C₁×TemporalDistance)^(ClearnessPower)  (44)where TemporalDistance=Distance (meters)/CloudSpeed (meters per second),ClearnessPower=ln(C₂Δt)−9.3, such that 5≤k≤15, where the expected valueis k=9.3, Δt is the desired output time interval (seconds),C₁=10⁻³seconds⁻¹, and C₂=1 seconds⁻¹.

Results also indicate that the correlation coefficient for the change inclearness index between two locations closely matches the valuescalculated using the following empirical relationship:ρ^(ΔKt) ^(i) ^(,ΔKt) ^(j) =(ρ^(Kt) ^(i) ^(,Kt) ^(j))^(ΔClearnessPower)  (45)where ρ^(Kt) ^(i) ^(,Kt) ^(j) is calculated using Equation (44) and

${{\Delta\;{ClearnessPower}} = {1 + \frac{140}{C_{2}\Delta\; t}}},$such that 100≤m≤200, where the expected value is m=140.

Empirical results also lead to the following models that may be used totranslate the variance of clearness index and the variance of change inclearness index from the measured time interval (Δt ref) to the desiredoutput time interval (Δt).

$\begin{matrix}{\sigma_{{Kt}_{\Delta\; t}}^{2} = {\sigma_{{Kt}_{\Delta\; t\mspace{11mu}{ref}}}^{2}{\exp\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{11mu}{ref}} \right)^{C_{3}}} \right\rbrack}}} & (46) \\{\sigma_{\Delta\;{Kt}_{\Delta\; t}}^{2} = {\sigma_{\Delta\;{Kt}_{\Delta\; t\mspace{11mu}{ref}}}^{2}\left\{ {1 - {2\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{11mu}{ref}} \right)^{C_{3}}} \right\rbrack}} \right\}}} & (47)\end{matrix}$where C₃=0.1≤C₃≤0.2, where the expected value is C₃=0.15.

FIGS. 8A-8B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds. Forexample, if the variance is calculated at a 300-second time interval andthe user desires results at a 10-second time interval, the adjustmentfor the variance clearness index would be 1.49

These empirical models represent a valuable means to rapidly calculatecorrelation coefficients and translate time interval withreadily-available information, which avoids the use ofcomputation-intensive calculations and high-speed streams of data frommany point sources, as would otherwise be required.

Validation

Equations (44) and (45) were validated by calculating the correlationcoefficients for every pair of locations in the Cordelia Junctionnetwork and the Napa network at half-hour time periods. The correlationcoefficients for each time period were then weighted by thecorresponding variance of that location and time period to determineweighted average correlation coefficient for each location pair. Theweighting was performed as follows:

${\overset{\_}{\rho^{{Kt}^{i},{Kt}^{J}}} = \frac{\sum\limits_{t = 1}^{T}{\sigma_{{{Kt} - i},j_{t}}^{2}\rho_{t}^{{Kt}^{i},{Kt}^{j}}}}{\sum\limits_{t = 1}^{T}\sigma_{{{Kt} - i},j_{t}}^{2}}},{and}$$\overset{\_}{\rho^{{\Delta\;{Kt}^{i}},{\Delta\;{Kt}^{J}}}} = {\frac{{\sum\limits_{t = 1}^{T}\sigma_{{\Delta\;{Kt}} - i}^{2}},{j_{t}\rho_{t}^{{\Delta\;{Kt}^{i}},{\Delta\;{Kt}^{j}}}}}{\sum\limits_{i = 1}^{T}\sigma_{{{\Delta\;{Kt}} - i},j_{t}}^{2}}.}$

FIGS. 9A-9F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance. FIGS. 10A-10F are graphs depicting, by way ofexample, the same information as depicted in FIGS. 9A-9F versus temporaldistance, based on the assumption that cloud speed was 6 meters persecond. The upper line and dots appearing in close proximity to theupper line present the clearness index and the lower line and dotsappearing in close proximity to the lower line present the change inclearness index for time intervals from 10 seconds to 5 minutes. Thesymbols are the measured results and the lines are the predictedresults.

Several observations can be drawn based on the information provided bythe FIGS. 9A-9F and 10A-10F. First, for a given time interval, thecorrelation coefficients for both the clearness index and the change inthe clearness index follow an exponential decline pattern versusdistance (and temporal distance). Second, the predicted results are agood representation of the measured results for both the correlationcoefficients and the variances, even though the results are for twoseparate networks that vary in size by a factor of 100. Third, thechange in the clearness index correlation coefficient converges to theclearness correlation coefficient as the time interval increases. Thisconvergence is predicted based on the form of the empirical modelbecause ΔClearnessPower approaches one as Δt becomes large.

Equation (46) and (47) were validated by calculating the averagevariance of the clearness index and the variance of the change in theclearness index across the 25 locations in each network for everyhalf-hour time period. FIGS. 11A-11F are graphs depicting, by way ofexample, the predicted versus the measured variances of clearnessindexes using different reference time intervals. FIGS. 12A-12F aregraphs depicting, by way of example, the predicted versus the measuredvariances of change in clearness indexes using different reference timeintervals. FIGS. 11A-11F and 12A-12F suggest that the predicted resultsare similar to the measured results.

Discussion

The point-to-point correlation coefficients calculated using theempirical forms described supra refer to the locations of specificphotovoltaic power production sites. Importantly, note that the dataused to calculate these coefficients was not obtained from time sequencemeasurements taken at the points themselves. Rather, the coefficientswere calculated from fleet-level data (cloud speed), fixed fleet data(distances between points), and user-specified data (time interval).

The empirical relationships of the foregoing types of empiricalrelationships may be used to rapidly compute the coefficients that arethen used in the fundamental mathematical relationships. The methodologydoes not require that these specific empirical models be used andimproved models will become available in the future with additional dataand analysis.

EXAMPLE

This section provides a complete illustration of how to apply themethodology using data from the Napa network of 25 irradiance sensors onNov. 21, 2010. In this example, the sensors served as proxies for anactual 1 kW photovoltaic fleet spread evenly over the geographicalregion as defined by the sensors. For comparison purposes, a directmeasurement approach is used to determine the power of this fleet andthe change in power, which is accomplished by adding up the 10-secondoutput from each of the sensors and normalizing the output to a 1 kWsystem. FIGS. 13A-13F are graphs and a diagram depicting, by way ofexample, application of the methodology described herein to the Napanetwork.

The predicted behavior of the hypothetical photovoltaic fleet wasseparately estimated using the steps of the methodology described supra.The irradiance data was measured using ground-based sensors, althoughother sources of data could be used, including from existingphotovoltaic systems or satellite imagery. As shown in FIG. 13A, thedata was collected on a day with highly variable clouds with one-minuteglobal horizontal irradiance data collected at one of the 25 locationsfor the Napa network and specific 10-second measured power outputrepresented by a blue line. This irradiance data was then converted fromglobal horizontal irradiance to a clearness index. The mean clearnessindex, variance of clearness index, and variance of the change inclearness index was then calculated for every 15-minute period in theday. These calculations were performed for each of the 25 locations inthe network. Satellite-based data or a statistically-significant subsetof the ground measurement locations could have also served in place ofthe ground-based irradiance data. However, if the data had beencollected from satellite regions, an additional translation from areastatistics to average point statistics would have been required. Theaveraged irradiance statistics from Equations (1), (10) and (11) areshown in FIG. 13B, where standard deviation (σ) is presented, instead ofvariance (σ²) to plot each of these values in the same units.

In this example, the irradiance statistics need to be translated sincethe data were recorded at a time interval of 60 seconds, but the desiredresults are at a 10-second resolution. The translation was performedusing Equations (46) and (47) and the result is presented in FIG. 13C.

The details of the photovoltaic fleet configuration were then obtained.The layout of the fleet is presented in FIG. 13D. The details includethe location of the each photovoltaic system (latitude and longitude),photovoltaic system rating (1/25 kW), and system orientation (all arehorizontal).

Equation (43), and its associated component equations, were used togenerate the time series data for the photovoltaic fleet with theadditional specification of the specific empirical models, as describedin Equations (44) through (47). The resulting fleet power and change inpower is presented represented by the red lines in FIGS. 13E and 13F.

Probability Density Function

The conversion from area statistics to point statistics relied upon twoterms A_(Kt) and A_(ΔKt) to calculate σ_(Kt) ² and σ_(ΔKt) ²,respectively. This section considers these terms in more detail. Forsimplicity, the methodology supra applies to both Kt and ΔKt, so thisnotation is dropped. Understand that the correlation coefficient σi,jcould refer to either the correlation coefficient for clearness index orthe correlation coefficient for the change in clearness index, dependingupon context. Thus, the problem at hand is to evaluate the followingrelationship:

$\begin{matrix}{A = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}}} & (48)\end{matrix}$

The computational effort required to calculate the correlationcoefficient matrix can be substantial. For example, suppose that the onewants to evaluate variance of the sum of points within a 1 squarekilometer satellite region by breaking the region into one millionsquare meters (1,000 meters by 1,000 meters). The complete calculationof this matrix requires the examination of 1 trillion (10¹²) locationpair combinations.

Discrete Formulation

The calculation can be simplified using the observation that many of theterms in the correlation coefficient matrix are identical. For example,the covariance between any of the one million points and themselvesis 1. This observation can be used to show that, in the case of arectangular region that has dimension of H by W points (total of N) andthe capacity is equal distributed across all parts of the region that:

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right){\quad{{\left\lbrack {{\sum\limits_{i = 0}^{H - 1}{\sum\limits_{j = 0}^{i}{{2^{k}\left\lbrack {\left( {H - i} \right)\left( {W - j} \right)} \right\rbrack}\rho^{d}}}} + {\sum\limits_{i = 0}^{W - 1}{\sum\limits_{j = 0}^{i}{{2^{k}\left\lbrack {\left( {W - i} \right)\left( {H - j} \right)} \right\rbrack}\rho^{d}}}}} \right\rbrack\mspace{20mu}{where}\text{:}\mspace{20mu} k} = \begin{matrix}{{- 1},{{{when}\mspace{14mu} i} = {{0\mspace{14mu}{and}\mspace{14mu} j} = 0}}} \\{1,{{{when}\mspace{14mu} j} = {{0\mspace{14mu}{or}\mspace{14mu} j} = i}}} \\{2,{{{when}\mspace{14mu} 0} < j < i}}\end{matrix}}}}} & (49)\end{matrix}$

When the region is a square, a further simplification can be made.

$\begin{matrix}{{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = 0}^{i}{2^{k}\left( {\sqrt{N} - i} \right)\left( {\sqrt{N} - j} \right)\rho^{d}}}} \right\rbrack}}{{where}\text{:}}{{k = \begin{matrix}{0,{{{when}\mspace{14mu} i} = {{0\mspace{14mu}{and}\mspace{14mu} j} = 0}}} \\{2,{{{when}\mspace{14mu} j} = {{0\mspace{14mu}{or}\mspace{14mu} j} = i}}} \\{3,{{{when}\mspace{14mu} 0} < j < i}}\end{matrix}},{and}}\text{}{d = {\left( \sqrt{i^{2} + j^{2}} \right){\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right).}}}} & (50)\end{matrix}$The benefit of Equation (50) is that there are

$\frac{N - \sqrt{N}}{2}$rather than N² unique combinations that need to be evaluated. In theexample above, rather than requiring one trillion possible combinations,the calculation is reduced to one-half million possible combinations.

Continuous Formulation

Even given this simplification, however, the problem is stillcomputationally daunting, especially if the computation needs to beperformed repeatedly in the time series. Therefore, the problem can berestated as a continuous formulation in which case a proposedcorrelation function may be used to simplify the calculation. The onlyvariable that changes in the correlation coefficient between any of thelocation pairs is the distance between the two locations; all othervariables are the same for a given calculation. As a result, Equation(50) can be interpreted as the combination of two factors: theprobability density function for a given distance occurring and thecorrelation coefficient at the specific distance.

Consider the probability density function. The actual probability of agiven distance between two pairs occurring was calculated for a 1,000meter×1,000 meter grid in one square meter increments. The evaluation ofone trillion location pair combination possibilities was evaluated usingEquation (48) and by eliminating the correlation coefficient from theequation. FIG. 14 is a graph depicting, by way of example, an actualprobability distribution for a given distance between two pairs oflocations, as calculated for a 1,000 meter×1,000 meter grid in onesquare meter increments.

The probability distribution suggests that a continuous approach can betaken, where the goal is to find the probability density function basedon the distance, such that the integral of the probability densityfunction times the correlation coefficient function equals:A=∫f(D)ρ(d)dD  (51)

An analysis of the shape of the curve shown in FIG. 14 suggests that thedistribution can be approximated through the use of two probabilitydensity functions. The first probability density function is a quadraticfunction that is valid between 0 and √{square root over (Area)}.

$\begin{matrix}{f_{Quad} = \left\{ \begin{matrix}{\left( \frac{6}{Area} \right)\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)} & {{{for}\mspace{14mu} 0} \leq D \leq \sqrt{Area}} \\0 & {{{for}\mspace{14mu} D} > \sqrt{Area}}\end{matrix} \right.} & (52)\end{matrix}$

This function is a probability density function because integratingbetween 0 and √{square root over (Area)} equals 1, that is,P[0≤D≤√{square root over (Area)}]=∫₀^(√{square root over (Area)})f_(Quad.)dD=1.

The second function is a normal distribution with a mean of √{squareroot over (Area)} and standard deviation of 0.1√{square root over(Area)}.

$\begin{matrix}{f_{Norm} = {\left( \frac{1}{0.1*\sqrt{Area}} \right)\left( \frac{1}{\sqrt{2\pi}} \right)e^{{- {(\frac{1}{2})}}{(\frac{D - \sqrt{Area}}{0.1*\sqrt{Area}})}^{2}}}} & (53)\end{matrix}$

Likewise, integrating across all values equals 1.

To construct the desired probability density function, take, forinstance, 94 percent of the quadratic density function plus 6 of thenormal density function.f=0.94∫₀ ^(√{square root over (Area)}) f _(Quad) dD+0.06∫_(−∞) ^(+∞) f_(Norm) dD  (54)

FIG. 15 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

The result is that the correlation matrix of a square area with uniformpoint distribution as N gets large can be expressed as follows, firstdropping the subscript on the variance since this equation will work forboth Kt and ΔKt.A≈[0.94∫₀ ^(√{square root over (Area)}) f _(Quadρ)(D)dD+0.06∫_(−∞) ^(+∞)f _(Norm)ρ(D)dD]  (55)where ρ(D) is a function that expresses the correlation coefficient as afunction of distance (D).Area to Point Conversion Using Exponential Correlation Coefficient

Equation (55) simplifies the problem of calculating the correlationcoefficient and can be implemented numerically once the correlationcoefficient function is known. This section demonstrates how a closedform solution can be provided, if the functional form of the correlationcoefficient function is exponential.

Noting the empirical results as shown in the graph in FIGS. 9A-9F, anexponentially decaying function can be taken as a suitable form for thecorrelation coefficient function. Assume that the functional form ofcorrelation coefficient function equals:

$\begin{matrix}{{\rho(D)} = e^{\frac{x\mspace{11mu} D}{\sqrt{Area}}}} & (56)\end{matrix}$Let Quad be the solution to ∫₀^(√{square root over (Area)})f_(Quad.)ρ(D)dD.

$\begin{matrix}{{Quad} = {{\int_{0}^{\sqrt{Area}}{f_{Quad}{\rho(D)}\ d\; D}} = {\left( \frac{6}{Area} \right){\int_{0}^{\sqrt{Area}}{{\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)\left\lbrack e^{\frac{xD}{\;^{\sqrt{Area}}}} \right\rbrack}\ d\; D}}}}} & (57)\end{matrix}$

Integrate to solve.

$\begin{matrix}{{Quad} = {(6)\left\lbrack {{\left( {{\frac{x}{\sqrt{Area}}D} - 1} \right)e^{\frac{xD}{\sqrt{Area}}}} - {\left( {{\left( \frac{x}{\sqrt{Area}} \right)^{2}D^{2}} - {2\frac{x}{\sqrt{Area}}D} + 2} \right)e^{\frac{xD}{\sqrt{Area}}}}} \right\rbrack}} & (58)\end{matrix}$

Complete the result by evaluating at D equal to √{square root over(Area)} for the upper bound and 0 for the lower bound. The result is:

$\begin{matrix}{{Quad} = {\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} & (59)\end{matrix}$

Next, consider the solution to ∫_(−∞) ^(+∞)f_(Norm.)ρ(D)dD which will becalled Norm.

$\begin{matrix}{{Norm} = {\left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\;\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}{(\frac{D - \mu}{\sigma})}^{2}}e^{\frac{xD}{\sqrt{Area}}}\ d\; D}}}} & (60)\end{matrix}$where μ=√{square root over (Area)} and σ=0.1√{square root over (Area)}.Simplifying:

$\begin{matrix}{{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}} \right\rbrack\left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\;\pi}} \right){\int_{- \infty}^{+ \infty}{e^{- {{(\frac{1}{2})}\lbrack\frac{D - {({\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}})}}{\sigma}\rbrack}^{2}}\ d\; D}}}} & (61)\end{matrix}$Substitute

$z = \frac{D - \left( {\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}} \right)}{\sigma}$and σdz=dD.

$\begin{matrix}{{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}} \right\rbrack\left( \frac{1}{\sqrt{2\;\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}z^{2}}\ d\; z}}}} & (62)\end{matrix}$Integrate and solve.

$\begin{matrix}{{Norm} = e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}}} & (63)\end{matrix}$Substitute the mean of √{square root over (Area)} and the standarddeviation of 0.1√{square root over (Area)} into Equation (55).Norm=e^(×(1+0.005x))   (64)

Substitute the solutions for Quad and Norm back into Equation (55). Theresult is the ratio of the area variance to the average point variance.This ratio was referred to as A (with the appropriate subscripts andsuperscripts) supra.

$\begin{matrix}{A = {{0.94{\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} + {0.06e^{x{({1 + {0.005x}})}}}}} & (65)\end{matrix}$

EXAMPLE

This section illustrates how to calculate A for the clearness index fora satellite pixel that covers a geographical surface area of 1 km by 1km (total area of 1,000,000 m²), using a 60-second time interval, and 6meter per second cloud speed. Equation (56) required that thecorrelation coefficient be of the form

$e^{\frac{xD}{\sqrt{Area}}}.$The empirically derived result in Equation (44) can be rearranged andthe appropriate substitutions made to show that the correlationcoefficient of the clearness index equals

${\exp\left\lbrack \frac{\left( {{\ln\;\Delta\; t} - 9.3} \right)D}{1000\mspace{14mu}{CloudSpeed}} \right\rbrack}.$Multiply the exponent by

$\frac{\sqrt{Area}}{\sqrt{Area}},$so that the correlation coefficient equals

$\exp{\left\{ {\left\lbrack \frac{\left( {{\ln\;\Delta\; t} - 9.3} \right)\sqrt{Area}}{1000\mspace{14mu}{CloudSpeed}} \right\rbrack\left\lbrack \frac{D}{\sqrt{Area}} \right\rbrack} \right\}.}$This expression is now in the correct form to apply Equation (65), where

$x = {\frac{\left( {{\ln\;\Delta\; t} - 9.3} \right)\sqrt{Area}}{1000\mspace{14mu}{CloudSpeed}}.}$Inserting the assumptions results in

${x = {\frac{\left( {{\ln\mspace{11mu} 60} - 9.3} \right)\sqrt{1,000,000}}{1000 \times 6} = {- 0.86761}}},$which is applied to Equation (65). The result is that A equals 65percent, that is, the variance of the clearness index of the satellitedata collected over a 1 km² region corresponds to 65 percent of thevariance when measured at a specific point. A similar approach can beused to show that the A equals 27 percent for the change in clearnessindex. FIG. 16 is a graph depicting, by way of example, resultsgenerated by application of Equation (65).Time Lag Correlation Coefficient

This section presents an alternative approach to deriving the time lagcorrelation coefficient. The variance of the sum of the change in theclearness index equals:σ_(ΣΔKt) ² =VAR[Σ(Kt ^(Δt) −Kt)]  (66)where the summation is over N locations. This value and thecorresponding subscripts have been excluded for purposes of notationalsimplicity. Divide the summation into two parts and add severalconstants to the equation:

$\begin{matrix}{\sigma_{{\Sigma\Delta}\;{Kt}}^{2} = {{VAR}\left\lbrack {{\sigma_{\Sigma\;{Kt}^{\Delta\; t}}\left( \frac{\Sigma\;{Kt}^{\Delta\; t}}{\sigma_{\Sigma\;{Kt}^{\Delta\; t}}} \right)} - {\sigma_{\Sigma\;{Kt}}\left( \frac{\Sigma\;{Kt}}{\sigma_{\Sigma\;{Kt}}} \right)}} \right\rbrack}} & (67)\end{matrix}$Since σ_(ΣKt) _(Δt) ≈σ_(ΣKt) (or σ_(ΣKt) _(Δt) =σΣ_(Kt) if the firstterm in Kt and the last term in Kt^(Δt) are the same):

$\begin{matrix}{\sigma_{{\Sigma\Delta}\;{Kt}}^{2} = {\sigma_{\Sigma\;{Kt}}^{2}{{VAR}\left\lbrack {\frac{\Sigma\;{Kt}^{\Delta\; t}}{\sigma_{\Sigma\;{Kt}^{\Delta\; t}}} - \frac{\Sigma\;{Kt}}{\sigma_{\Sigma\;{Kt}}}} \right\rbrack}}} & (68)\end{matrix}$The variance term can be expanded as follows:

$\begin{matrix}{\sigma_{\Sigma\;\Delta\;{Kt}}^{2} = {\sigma_{\Sigma\;{Kt}}^{2}\left\{ {\frac{{VAR}\left\lbrack {\Sigma\;{Kt}^{\Delta\; t}} \right\rbrack}{\sigma_{\Sigma\;{Kt}^{\Delta\; t}}^{2}} + \frac{{VAR}\left\lbrack {\Sigma\;{Kt}} \right\rbrack}{\sigma_{\Sigma\;{Kt}}^{2}} - \frac{2{{COV}\left\lbrack {{\Sigma\;{Kt}},{\Sigma\;{Kt}^{\Delta\; t}}} \right\rbrack}}{\sigma_{\Sigma\;{Kt}}\sigma_{\Sigma\;{Kt}^{\Delta\; t}}}} \right\}}} & (69)\end{matrix}$Since COV[ΣKt,ΣKt^(Δt)]=σ_(ΣKt)σ_(ΣKt) _(Δt) ρ^(ΣKt,ΣKt) ^(Δt) , thefirst two terms equal one and the covariance term is replaced by thecorrelation coefficient.σ_(ΣΔKt) ²=σ_(ΣKt) ²(1−ρ^(ΣKt,ΣKt) ^(ΔT) )   (70)This expression rearranges to:

$\begin{matrix}{\rho^{{\Sigma\;{Kt}},{\Sigma\;{Kt}^{\Delta\; t}}} = {1 - {\frac{1}{2}\frac{\sigma_{\Sigma\;\Delta\;{Kt}}^{2}}{\sigma_{\Sigma\;{Kt}}^{2}}}}} & (71)\end{matrix}$

Assume that all photovoltaic plant ratings, orientations, and areaadjustments equal to one, calculate statistics for the clearness aloneusing the equations described supra and then substitute. The result is:

$\begin{matrix}{\rho^{{\Sigma\;{Kt}},{\Sigma\;{Kt}^{\Delta\; t}}} = {1 - {\frac{P^{\Delta\;{Kt}}}{2P^{Kt}}\frac{\sigma_{\;\overset{\_}{\Delta\;{Kt}}}^{2}}{\sigma_{\overset{\_}{Kt}}^{2}}}}} & (72)\end{matrix}$Relationship Between Time Lag Correlation Coefficient and Power/Changein Power Correlation Coefficient

This section derives the relationship between the time lag correlationcoefficient and the correlation between the series and the change in theseries for a single location.

$\rho^{P,{\Delta\; P}} = {\frac{{COV}\left\lbrack {P,{\Delta\; P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}} = {\frac{{COV}\left\lbrack {P,{P^{\Delta\; t} - P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}} = \frac{{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}}}}$Since  σ_(Δ P)² = VAR[P^(Δ t) − P] = σ_(P)² + σ_(P^(Δ t))² − 2COV[P, P^(Δ t)]  and${{{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack} = {\rho^{P,{P\;\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}},{then}}\mspace{14mu}$$\rho^{P,{\Delta\; P}} = {\frac{{\rho^{P,{P\;\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\left( {\sigma_{P}^{2} + \sigma_{P^{\Delta\; t}}^{2} - {2\rho^{P,P^{\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}} \right)}}.}$Since σ_(P) ²≈σ_(P) _(Δt) ², this expression can be further simplified.Then, square both expression and solve for the time lag correlationcoefficient:ρ^(P,P) ^(Δt) =1−2(ρ^(P,ΔP))²Correlation Coefficients Between Two Regions

Assume that the two regions are squares of the same size, each side withN points, that is, a matrix with dimensions of √{square root over (N)}by √{square root over (N)} points, where √{square root over (N)} is aninteger, but are separated by one or more regions. Thus:

$\begin{matrix}{{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( \frac{1}{N^{2}} \right)\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = {1 - \sqrt{N}}}^{\sqrt{N} - 1}{{k\left( {\sqrt{N} - i} \right)}\left( {\sqrt{N} - {j}} \right)\rho^{d}}}} \right\rbrack}} & (73)\end{matrix}$where

$k = \left\{ {\begin{matrix}1 & {{{when}\mspace{14mu} i} = 0} \\2 & {{{when}\mspace{14mu} i} > 0}\end{matrix},{d = {\left( \sqrt{i^{2} + \left( {j + {M\sqrt{N}}} \right)^{2}} \right)\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right)}},} \right.$and M equals the number of regions.

FIG. 17 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions. FIG. 17suggests that the probability density function can be estimated usingthe following distribution:

$\begin{matrix}{f = \left\{ \begin{matrix}{1 - \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{{for}\mspace{14mu}{Spacing}} - \sqrt{Area}} \leq D \leq {Spacing}} \\{1 + \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{for}\mspace{14mu}{Spacing}} \leq D \leq {{Spacing} + \sqrt{Area}}} \\0 & {{all}\mspace{14mu}{else}}\end{matrix} \right.} & (74)\end{matrix}$

This function is a probability density function because the integrationover all possible values equals zero. FIG. 18 is a graph depicting, byway of example, results by application of this model.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

What is claimed is:
 1. A method for managing photovoltaic fleetproduction with the aid of a digital computer, comprising the steps of:regularly measuring a set of irradiance observations with an irradiancesensor for a first physical location in a geographic region within whicha photovoltaic power production fleet is situated, the photovoltaicpower production fleet being connected to a power grid and comprising aplurality of photovoltaic power production plants deployed at differentphysical locations within the geographic region, the power grid furthercomprising a plurality of power generators other than the photovoltaicpower production plants in the photovoltaic fleet, a transmissioninfrastructure, and a power distribution infrastructure for distributingpower from the photovoltaic system and the power generators toconsumers; regularly measuring cloud speed with a wind sensor for thefirst physical location; and centrally operating with a digital computercomprising steps of: selecting a second physical location in thegeographic region that is different than the first physical location atwhich one of the photovoltaic power production plants is located;regularly providing to the computer the set of irradiance observationsfrom the irradiance sensor and the cloud speed from the wind sensor,plus clear sky global horizontal irradiance for the geographic region;determining, by the computer, a set of input sky clearness indexes as aratio of each of the irradiance observations in the set of irradianceobservations and the clear sky global horizontal irradiance;determining, by the computer, a temporal distance as a physical distancebetween the first and the second physical locations in proportion to thecloud speed; calculating, by the computer, a clearness index correlationcoefficient between the first and the second physical locations as anempirically-derived function of the temporal distance and boundingstatistical error between the first and the second physical locations byevaluating a mean and standard deviation; weighting, by the computer,the set of input sky clearness indexes by the clearness indexcorrelation coefficient to form a set of output sky clearness indexes,which indicates the sky clearness for the second physical location, andproportioning the mean and standard deviation to the output set of skyclearness indexes; regularly estimating photovoltaic power production ofthe photovoltaic power production plant located at the second physicallocation in proportion to the output set of clear sky indexes as boundedby the proportioned mean and standard deviation; and regularly adjustinga forecast of aggregate photovoltaic power production for thephotovoltaic power production fleet comprising the estimated powerproduction of the photovoltaic power production plant located at thesecond physical location and estimated power production of the otherphotovoltaic power production plants comprised in the photovoltaic powerproduction fleet.
 2. A method according to claim 1, further comprisingthe steps of at least one of: evaluating, by the computer, a variance ofthe output set of sky clearness indexes; and evaluating in the computera change in the variance of the output set of sky clearness indexes. 3.A method according to claim 2, wherein the variance σ_(Kt) ² of the skyclearness index Kt from a measured time interval Δt ref to a desiredoutput time interval Δt is determined in accordance with:$\sigma_{{Kt}_{\Delta\; t}}^{2} = {\sigma_{{Kt}_{\Delta\; t\;{ref}}}^{2}{\exp\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{14mu}{ref}} \right)^{C_{1}}} \right\rbrack}}$where 0.1≤C₁≤0.2.
 4. A method according to claim 2, wherein the varianceσ_(ΔKt) ² of the change in the sky clearness index ΔKt from a measuredtime interval Δt ref to a desired output time interval Δt is determinedin accordance with:$\sigma_{{\Delta{Kt}}_{\Delta\; t}}^{2} = {\sigma_{{\Delta Kt}_{\Delta{tref}}\mspace{11mu}}^{2}\left\{ {1 - {2\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{14mu}{ref}} \right)^{C_{1}}} \right\rbrack}} \right\}}$where 0.1≤C₁≤0.2.
 5. A method according to claim 1, further comprisingthe steps of: selecting, by the computer, a time period comprising atime resolution by which the each irradiance observations in the set ofirradiance observations was regularly measured; dividing, by thecomputer, the set of irradiance observations by increments of the timeperiod into a time series set of irradiance observations; and forming,by the computer, a time series set of the output sky clearness indexesover each time period increment in the time series set of irradianceobservations and bounding statistical error at each physical location inthe geographic region by evaluating a mean and standard deviation.
 6. Amethod according to claim 5, further comprising the steps of forming, bythe computer, a full set of the output sky clearness indexes for aplurality of additional physical locations, which are each within thegeographic region; forming, by the computer, a full time series set ofthe full set of the output sky clearness indexes over each time periodincrement in the time series; determining, by the computer, irradiancestatistics from the mean and standard deviation at each additionalphysical location in the geographic region for the full time series setof the output sky clearness indexes; and building, by the computer,power statistics for the photovoltaic fleet as a function of the fleetirradiance statistics and a power rating of the photovoltaic fleet.
 7. Amethod according to claim 6, further comprising the step of: generating,by the computer, a time series of the power statistics for thephotovoltaic fleet by applying a time lag correlation coefficient for anoutput time interval to the power statistics over each of the timeperiod increments.
 8. A method according to claim 1, wherein theclearness index correlation coefficient ρ^(Kt) ^(i) ^(,Kt) ^(j) for thefirst and the second physical locations i and j is determined inaccordance with:ρ^(Kt) ^(i) ^(,Kt) ^(j) =exp(C ₁×TemporalDistance)^(ClearnessPower)where C₁ equals 10⁻³ seconds⁻¹ and:Clearness Power=In(C ₂ Δt)−k such that Δt is a measured time interval inseconds, C₂ equals 1 seconds⁻¹, and 5 ≤k≤15.
 9. A method according toclaim 8 wherein the change in the clearness index correlationcoefficientρ^(ΔKt) ^(i) ^(,ΔKt) ^(j) for the first and the secondphysical locations i and j is determined in accordance with:ρ^(ΔKt) ^(i) ^(,ΔKt) ^(j) =(ρ^(Kt) ^(i) ^(,Kt) ^(j) )^(ΔClearnessPower)where: ${\Delta\;{ClearnessPower}} = {1 + \frac{m}{C_{2}\Delta\; t}}$such that 100 ≤m≤200.
 10. A system for managing photovoltaic fleetproduction with the aid of a digital computer, comprising: an irradiancesensor for a first physical location in a geographic region within whicha photovoltaic power production fleet comprised in a power grid issituated, the photovoltaic power production fleet comprising a pluralityof photovoltaic power production plants deployed at different physicallocations within the geographic region, the irradiance sensor configuredto regularly measure a set of irradiance observations, the power gridfurther comprising a plurality of power generators other than thephotovoltaic power production plants in the photovoltaic fleet, atransmission infrastructure, and a power distribution infrastructure fordistributing power from the photovoltaic system and the power generatorsto consumers; a wind sensor for the first physical location, the windsensor configured to regularly measure cloud speed; a non-transitorycomputer readable storage medium comprising program code; and a computerprocessor and memory on a digital computer configured to centrallyoperate the photovoltaic power production fleet with the computerprocessor coupled to the storage medium, wherein the computer processoris configured to execute the program code to perform steps to: select asecond physical location in the geographic region that is different thanthe first physical location at which one of the photovoltaic powerproduction plants is located; regularly provide the set of irradianceobservations from the irradiance sensor and the cloud speed from thewind sensor, plus clear sky global horizontal irradiance for thegeographic region; determine a set of input sky clearness indexes as aratio of each of the irradiance observations in the set of irradianceobservations and the clear sky global horizontal irradiance; determine atemporal distance as a physical distance between the first and thesecond physical locations in proportion to the cloud speed; calculate aclearness index correlation coefficient between the first and the secondphysical locations as an empirically-derived function of the temporaldistance and bounding statistical error between the first and the secondphysical locations by evaluating a mean and standard deviation; weightthe set of input sky clearness indexes by the clearness indexcorrelation coefficient to form a set of output sky clearness indexes,which indicates the sky clearness for the second physical location, andproportioning the mean and standard deviation to the output set of skyclearness indexes; and regularly estimate photovoltaic power productionof the photovoltaic power production plant located at the secondphysical location in proportion to the output set of clear sky indexesas bounded by the proportioned mean and standard deviation; andregularly adjust a forecast of aggregate photovoltaic power productionfor the photovoltaic power production fleet comprising the estimatedpower production of the photovoltaic power production plant located atthe second physical location and estimated power production of the otherphotovoltaic power production plants comprised in the photovoltaic powerproduction fleet.
 11. A system according to claim 10, wherein thecomputer processor is further configured to execute the program code toperform at least one of the steps to: evaluate a variance of the outputset of sky clearness indexes; and evaluate a change in the variance ofthe output set of sky clearness indexes.
 12. A system according to claim11, wherein the variance σ_(Kt) ² of the sky clearness index Kt from ameasured time interval ^(Δt ref) to a desired output time interval Δt isdetermined in accordance with:$\sigma_{{Kt}_{\Delta\; t}}^{2} = {\sigma_{{Kt}_{\Delta\; t\;{ref}}}^{2}{\exp\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{14mu}{ref}} \right)^{C_{1}}} \right\rbrack}}$where 0.1≤C₁≤0.2.
 13. A system according to claim 11, wherein thevariance σ_(ΔKt) ²of the change in the sky clearness index ^(ΔKt) from ameasured time interval Δt ref to a desired output time interval Δt isdetermined in accordance with:$\sigma_{\Delta\;{Kt}_{\Delta\; t}}^{2} = {\sigma_{{\Delta\;{Kt}_{\Delta\;{tref}}}\;}^{2}\left\{ {1 - {2\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{14mu}{ref}} \right)^{C_{1}}} \right\rbrack}} \right\}}$where 0.1≤C₁≤0.2.
 14. A system according to claim 10, wherein thecomputer processor is further configured to execute the program code toperform the steps to: select a time period comprising a time resolutionby which the each irradiance observations in the set of irradianceobservations was regularly measured; divide the set of irradianceobservations by increments of the time period into a time series set ofirradiance observations; and form a time series set of the output skyclearness indexes over each time period increment in the time series setof irradiance observations and bound statistical error at each physicallocation in the geographic region by evaluating a mean and standarddeviation.
 15. A system according to claim 14, wherein the computerprocessor is further configured to execute the program code to performthe steps to: form a full set of the output sky clearness indexes for aplurality of additional physical locations, which are each within thegeographic region; form a full time series set of the full set of theoutput sky clearness indexes over each time period increment in the timeseries; and determine irradiance statistics from the mean and standarddeviation at each additional physical location in the geographic regionfor the full time series set of the output sky clearness indexes, andbuild power statistics for the photovoltaic fleet as a function of thefleet irradiance statistics and a power rating of the photovoltaicfleet.
 16. A system according to claim 15, wherein the computerprocessor is further configured to execute the program code to performthe steps to: generate a time series of the power statistics for thephotovoltaic fleet by applying a time lag correlation coefficient for anoutput time interval to the power statistics over each of the timeperiod increments.
 17. A system according to claim 16, wherein theproduction output controller for the photovoltaic fleet is furtheradapted to receive the time series of the power statistics for thephotovoltaic fleet to a production output controller for thephotovoltaic fleet and operate the photovoltaic fleet to generatesufficient power to realize the time series of the power statistics. 18.A system according to claim 10, wherein the clearness index correlationcoefficient ρ^(Kt) ^(i) ^(,Kt) ^(j) for the first and the secondphysical locations i and j is determined in accordance with:ρ^(Kt) ^(i) ^(,Kt) ^(j) =exp(C ₁×TemporalDistance)^(ClearnessPower)where C₁ equals 10⁻³ seconds⁻¹ and:Clearness Power =In(C ₂ Δt)−k such that Δt is a measured time intervalin seconds, C₂ equals 1 seconds⁻¹, and 5 ≤k≤15.
 19. A system accordingto claim 18 wherein the change in the clearness index correlationcoefficient ρ^(ΔKt) ^(i) ^(,ΔKt) ^(j) for the first and the secondphysical locations i and j is determined in accordance with:ρ^(ΔKt) ^(i) ^(,ΔKt) ^(j) =(ρ^(Kt) ^(i) ^(,Kt) ^(j) )^(ΔClearnessPower)where: ${\Delta\;{ClearnessPower}} = {1 + \frac{m}{C_{2}\Delta\; t}}$such that 100≤m≤200.